Coexistence Of Periodic Solutions Research Articles

Multiple solutions and transient chaos in a nonlinear flexible coupling model

This paper investigates the nonlinear dynamics of a flexible tyre coupling via computer modelling and simulation. The research mainly focused on identifying basins of attraction of coexisting solutions of the formulated phenomenological coupling model. On the basis of the derived mathematical model, and by assuming ranges of variability of the control parameters, the areas in which chaotic clutch movement takes place are determined. To identify multiple solutions, a new diagram of solutions (DS) was used, illustrating the number of coexisting solutions and their periodicity. The DS diagram was drawn based on the fixed points of the Poincaré section. To verify the proposed method of identifying periodic solutions, the graphic image of the DS was compared to the three-dimensional distribution of the largest Lyapunov exponent and the bifurcation diagram. For selected values of the control parameter ω, coexisting periodic solutions were identified, and basins of attraction were plotted. Basins of attraction were determined in relation to examples of coexistence of periodic solutions and transient chaos. Areas of initial conditions that correspond to the phenomenon of unstable chaos are mixed with the conditions of a stable periodic solution, to which the transient chaos is attracted. In the graphic images of the basins of attraction, the areas corresponding to the transient and periodic chaos are blurred.

Coexistence of periodic solutions with various periods of impulsive differential equations and inclusions on tori via Poincaré operators

The coexistence of subharmonic periodic solutions of various orders is investigated to the first-order vector system of impulsive (upper-) Carathéodory differential equations and inclusions on tori. As the main tool, our recent Sharkovsky-type results for multivalued maps on tori are applied via the associated Poincaré translation operators along the trajectories of given systems. The solvability criteria are formulated, under natural bi-periodicity assumptions imposed on the right-hand sides, in terms of the Lefschetz numbers of admissible impulsive maps. Since the criteria become effective on the circle, the main general theorem can be improved and reformulated there in a more transparent way. The obtained results can be regarded in a certain sense as a nontrivial extension of those due to Poincaré [28], Denjoy [17] and van Kampen [24].

Global attractivity of the predator–prey system for cotton aphid and seven-spot ladybird beetle with stage structure

It is well known that the cotton aphid is the major pest in cotton fields of Northwest China, and seven-spot ladybird is an important natural enemy among the various possible natural enemies of cotton aphid. In order to increase the applications of population dynamics in integrated pest management and control the cotton aphids biologically, we need to understand the population dynamics of cotton aphid and their natural enemies. A delay predator–prey system on cotton aphid and seven-spot ladybird beetle are proposed in this paper. Based on the comparison theorem and an iterative method, we investigate the global attractivity of the equilibrium points which have important biological meanings. Furthermore, some numerical simulations were carried out to illustrate and expand our theoretical results, in which a conjecture to generalize the well-known Theorem 16.4 in H. R. Thiemes book was put forward, which was taken as the open problem. The numerical simulations show coexistence of periodic solution, confirming the theoretical prediction.

Computing and Controlling Basins of Attraction in Multistability Scenarios

The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps.

Spatially heterogeneous periodic solutions of the Hutchinson equation with distributed saturation

Asymptotics of spatially heterogeneous periodic solutions of the spatially distributed Hutchinson equation with periodic boundary conditions are obtained in cases of symmetric and asymmetric saturations. The numerical analysis of a simplified model shows the multistability, i.e., the coexistence of periodic solutions in the form of travelling waves, among which there are no solutions with the obtained asymptotics.

A drifting impact oscillator with periodic impulsive loading: Application to percussive drilling

Percussive drilling is extensively used to drill hard rocks in the earth resource industry, where it performs best compared to other drilling technologies. In this paper, we propose a novel model of the process that consists of a drifting oscillator under impulsive loading coupled with a bilinear force/penetration interface law, together with a kinetic energy threshold for continuous bit penetration. Following the formulation of the model, we analyze its steady-state response and show that there exists a parallel between theoretical and experimental predictions, as both exhibit a maximum of the average penetration rate with respect to the vertical load on bit. In addition, the existence of complex long-term dynamics with the coexistence of periodic solutions in certain parameter ranges is demonstrated.

Coexistence of periodic solutions of Ince's equation

Coexistence of periodic solutions of Ince's equation

On the coexistence of periodic solutions

(p(t; A)) wYyldt2 = > 5 = Pk 4Y, where X is a real parameter, and, for every /\, p(t; h) are real continuous periodic functions with period n defined for all t E (---co, 03). The problem of the coexistence of periodic solutions consists of studying such values of parameter h for which two linearly independent and periodic or hay-periodic1 solutions with period ?T of (p(t; h)) coexist. A detailed study of the problem can be found in W. Magnus’ and S. Winkler’s book on Hill’s Equations [2]. The first type is of the form